To what plane curve does a Catmull-Clark subdivision converge?
According to the original paper CATMULL, E., AND CLARK, J. Recursively generated B-spline surfaces on arbitrary topological
meshes. Computer-Aided Design 10 (Sept. 1978), 350–355.

For rectangular control-point meshes the Catmull Clark subdivision modifier creates a standard B-spline surface. Regardless of the number of subdivision it will not converge to an accurate circle, but with more control points the difference between a circle and the interpolated shape will be less noticeable.